{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true, "scrolled": false }, "outputs": [], "source": [ "using ApproxFun, Plots, ComplexPhasePortrait, ApproxFun, SingularIntegralEquations,\n", " SpecialFunctions, DualNumbers, SO\n", "\n", "using SingularIntegralEquations.HypergeometricFunctions\n", "gr();" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# M3M6: Methods of Mathematical Physics\n", "\n", "$$\n", "\\def\\dashint{{\\int\\!\\!\\!\\!\\!\\!-\\,}}\n", "\\def\\infdashint{\\dashint_{\\!\\!\\!-\\infty}^{\\,\\infty}}\n", "\\def\\D{\\,{\\rm d}}\n", "\\def\\E{{\\rm e}}\n", "\\def\\dx{\\D x}\n", "\\def\\dt{\\D t}\n", "\\def\\dz{\\D z}\n", "\\def\\ds{\\D s}\n", "\\def\\C{{\\mathbb C}}\n", "\\def\\R{{\\mathbb R}}\n", "\\def\\H{{\\mathbb H}}\n", "\\def\\CC{{\\cal C}}\n", "\\def\\HH{{\\cal H}}\n", "\\def\\FF{{\\cal F}}\n", "\\def\\I{{\\rm i}}\n", "\\def\\Ei{{\\rm Ei}\\,}\n", "\\def\\qqqquad{\\qquad\\qquad}\n", "\\def\\qqand{\\qquad\\hbox{and}\\qquad}\n", "\\def\\qqfor{\\qquad\\hbox{for}\\qquad}\n", "\\def\\qqwhere{\\qquad\\hbox{where}\\qquad}\n", "\\def\\Res_#1{\\underset{#1}{\\rm Res}}\\,\n", "\\def\\sech{{\\rm sech}\\,}\n", "\\def\\acos{\\,{\\rm acos}\\,}\n", "\\def\\erfc{\\,{\\rm erfc}\\,}\n", "\\def\\vc#1{{\\mathbf #1}}\n", "\\def\\ip<#1,#2>{\\left\\langle#1,#2\\right\\rangle}\n", "\\def\\br[#1]{\\left[#1\\right]}\n", "\\def\\norm#1{\\left\\|#1\\right\\|}\n", "\\def\\half{{1 \\over 2}}\n", "\\def\\fL{f_{\\rm L}}\n", "\\def\\fR{f_{\\rm R}}\n", "\\def\\HF(#1,#2,#3,#4){{}_2F_1\\!\\left({#1,#2 \\atop #3}; #4 \\right)}\n", "\\def\\questionequals{= \\!\\!\\!\\!\\!\\!{\\scriptstyle ? \\atop }\\,\\,\\,}\n", "$$\n", "\n", "Dr Sheehan Olver\n", "<br>\n", "s.olver@imperial.ac.uk\n", "\n", "Office Hours: 3-4pm Mondays, 11-12am Thursdays, Huxley 6M40\n", "<br>\n", "Website: https://github.com/dlfivefifty/M3M6LectureNotes\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "# Lecture 27: Singular points of ordinary differential equations\n", "\n", "\n", "2. Singular points of ODEs\n", " - regular singular point\n", " - irregular singular point\n", " - singular points at ∞" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Singular points\n", "\n", "We will consider the first order ODE\n", "$$\n", "u' + a(z) u = 0\n", "$$\n", "and the second order ODE\n", "$$\n", "u'' + a(z) u' + b(z) u = 0\n", "$$\n", "where $a$ and $b$ are rational. \n", "\n", "The simplest point is an ordinary point, where we know that $u$ is analytic:\n", "\n", "**Definition (ordinary point)** $z_0$ is a ordinary point of the ODE if $a(z)$ and $b(z)$ are analytic at $z_0$.\n", "\n", "\n", "### Regular singular point\n", "\n", "Suppose $a$ has a simple pole, and w.l.o.g. take $z_0=0$, that is: \n", "$$\n", "a(z) = {a_{-1} \\over z} + a_0 + a_1 z + a_2 z^2 + \\cdots \n", "$$\n", "where $a_0 + a_1 z + a_2 z^2 + \\cdots $ is analytic.\n", "Take as an ansatz to the first order ODE\n", "$$u(z) = z^\\alpha ( 1 + z v(z)) = z^\\alpha + z^{\\alpha+1} v$$\n", "where $\\alpha = a_{-1}$. Note that $c u(z)$ is a solution of $u'(z) = a(z) u(z)$: the choice of $1$ above makes the solution unique. \n", "\n", "\n", "Note that\n", "$$\n", "u'(z) = \\alpha z^{\\alpha-1} + z^\\alpha ((\\alpha+1) v + z v')\n", "$$\n", "The ODE $u'(z) = a(z) u(z)$ becomes an inhomogenous equation:\n", "\\begin{align*}\n", "z^\\alpha ((\\alpha+1) v + z v' - z a(z) v) &= z^{\\alpha} (a(z) - {\\alpha \\over z} ) & \\Leftrightarrow\\\\\n", "z v'(z) + (\\alpha+1 - z a(z)) v(z) &= a_0 + a_1 z + a_2 z^2 + \\cdots\n", "\\end{align*}\n", "In operator form, expressing \n", "$$\n", "v(z) = v_0 + v_1z + v_2 z^2 = (1,z,z^2,\\ldots)\\begin{pmatrix}v_0\\\\v_1\\\\\\vdots \\end{pmatrix}\n", "$$\n", "we have\n", "\\begin{align*}\n", "z v' &=(1,z,z^2,\\ldots) \\begin{pmatrix} 0 \\cr & 1 \\cr && 2 \\cr &&& 3 \\cr &&&&\\ddots \\end{pmatrix} \\begin{pmatrix}v_0\\\\v_1\\\\\\vdots \\end{pmatrix} \\\\\n", "(\\alpha+1 - z a(z)) v(z) &= (1,z,z^2,\\ldots) \\begin{pmatrix} 1 \\cr -a_0 & 1 \\cr -a_1 & -a_0& 1 \\cr -a_2 & -a_1&-a_0& 1 \\cr \\vdots&\\ddots&\\ddots&\\ddots&\\ddots \\end{pmatrix} \\begin{pmatrix}v_0\\\\v_1\\\\\\vdots \\end{pmatrix}\n", "\\end{align*}\n", "Thus the equation for our unknown coefficients becomes:\n", "$$\n", "\\begin{pmatrix} 1 \\cr -a_0 & 2 \\cr -a_1 & -a_0& 3 \\cr -a_2 & -a_1&-a_0& 4 \\cr \\vdots&\\ddots&\\ddots&\\ddots&\\ddots \\end{pmatrix} \\begin{pmatrix}v_0\\\\v_1\\\\\\vdots \\end{pmatrix} = \\begin{pmatrix}a_0\\\\a_1\\\\\\vdots \\end{pmatrix}\n", "$$\n", "This equation induces decay in $|v_k|$ just like last lecture, hence we know its analytic with the same radius of convergence as $a$.\n", "\n", "Here is an example: as we can see, `v(z)` is analytic in a disk of radius `r` where `r` is dictated by the function `a`. 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transform=\"translate(39, 12)\"/>\n", "</g>\n", "</svg>\n" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "r = 0.9 # radius we are solving in, can be up to but not including 1\n", "S = Taylor(Circle(r))\n", "α = 0.3\n", "a = z -> α/z + 1/(1-z)\n", "ã = Fun( z -> a(z) - α/z, S)\n", "za = Fun( z -> z*a(z), S)\n", "z = Fun(S)\n", "D = Derivative() : S → S\n", "L = z*D + (α+1-za)\n", "v = L \\ ã\n", "\n", "# v is analytic in disk of radius r\n", "phaseplot(v, (-1,1), (-1,1))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here is a plot of `u`, which shows that it has a branch cut:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n", "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 600 400\">\n", "<defs>\n", " <clipPath id=\"clip00\">\n", " <rect 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"source": [ "# u has a branch \n", "u = z -> z^α + z^(α+1)*v(z)\n", "phaseplot(u, (-1,1), (-1,1))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can confirm it solves the ODE:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "4.884981308350689e-15 - 7.521914234669674e-17im" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "up = dualpart(u(dual(0.1,1.0))) # uses DualNumbers.jl to calculate derivative\n", "\n", "up - a(0.1)u(0.1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Definition (first order regular singular point)** $z_0$ is a regular singular point of the first order ODE if $(z-z_0)a(z)$ is analytic at $z_0$.\n", "\n", "For second order equations, we will arrive at the following analogue:\n", "\n", "**Definition (second order regular singular point)** $z_0$ is a regular singular point of the second order ODE if $(z-z_0) a(z)$ and $(z-z_0)^2b(z)$ are analytic at $z_0$.\n", "\n", "Assume \n", "$$\n", "a(z) = {a_{-1} \\over z} + a_0 + a_1 z + \\cdots\n", "$$\n", "and\n", "$$\n", "b(z) = {b_{-2} \\over z^2} + {b_{-1} \\over z} + b_0 + b_1 z + \\cdots\n", "$$\n", "To determine the right ansatz, consider\n", "$$\n", "u'' + {a_{-1} \\over z} u' + {b_{-2} \\over z^2} u'' = 0\n", "$$\n", "which has solutions $z^\\alpha$ solving the indicial equation:\n", "\n", "**Definition (indicial equation)**\n", "$$\n", "\\alpha(\\alpha-1) + a_{-1} \\alpha + b_{-2} = 0\n", "$$\n", "\n", "Now take as an ansatz:\n", "$$\n", " u(z) = z^\\alpha (1 + z v(z)) = z^\\alpha + z^{\\alpha+1} v(z)\n", "$$\n", "Differentiating we have\n", "\\begin{align*}\n", "u'(z) &= \\alpha z^{\\alpha-1} + z^\\alpha((\\alpha+1) v + z v') \\\\\n", "u''(z) &= \\alpha(\\alpha-1) z^{\\alpha-2} + z^{\\alpha-1}( \\alpha(\\alpha+1) v + \\alpha z v' + (\\alpha+1) z v' + z v' + z^2 v'' ) \\\\\n", " &= \\alpha(\\alpha-1) z^{\\alpha-2} + z^{\\alpha-1}( \\alpha(\\alpha+1) v + 2(\\alpha+1) z v' + z^2 v'' ) \n", "\\end{align*}\n", "Thus $v$ solves \n", "$$\n", "z^2 v''+ z (2(\\alpha+1) + z a(z)) v' + ( \\alpha(\\alpha+1) + z a(z) (\\alpha+1) + z^2 b(z)) v = - {\\alpha(\\alpha-1) \\over z} - a(z) \\alpha -z b(z) = \\underbrace{- \\alpha a_0 - b_{-1} - (\\alpha a_1 + b_0) z - (\\alpha a_2 + b_1) z^2 + \\cdots}_{f(z) = f_0 + f_1 z + \\cdots}\n", "$$\n", "\n", "The question is when can forward substitution proceed. Let's write out the operators, first we have:\n", "$$\n", "z^2 v'' \\equiv \\begin{pmatrix} 0 \\cr & 0 \\cr &&2 \\cr &&& 6 \\cr &&&&12 \\cr &&&&&\\ddots \\end{pmatrix}\n", "$$\n", "Then we have\n", "$$\n", "2 (\\alpha+1) z v' \\equiv \\begin{pmatrix} 0\\cr & 2 (\\alpha+1) \\cr &&&4 (\\alpha+1) \\cr &&&& 6(\\alpha+1) \\cr &&&&&8 (\\alpha+1) \\cr &&&&&&\\ddots \\end{pmatrix}\n", "$$\n", "and\n", "$$\n", "z^2 a(z) v' = (z a(z))(z v') \\equiv \\begin{pmatrix} a_{-1} \\\\ a_0 & a_{-1} \\\\ a_1 & a_0 & a_{-1} \\\\ a_2 & a_1 & a_0 & a_{-1} \\\\ \\vdots &\\ddots&\\ddots&\\ddots&\\ddots \\end{pmatrix} \\begin{pmatrix} 0 \\cr & 1 \\cr &&2 \\cr &&& 3 \\cr &&&&4 \\cr &&&&&\\ddots \\end{pmatrix}\n", "$$\n", "Finally since\n", "\\begin{align*}\n", "\\alpha(\\alpha+1) + z (\\alpha+1) a(z) + z^2 b(z) &= \\alpha(\\alpha-1) + \\alpha a_{-1} + b_{-2} + 2 \\alpha + a_{-1} + ((\\alpha+1) a_0 + b_{-1}) z + ((\\alpha+2) a_1 + b_0) z^2 + \\cdots \\\\\n", "& = 2 \\alpha + a_{-1} + ((\\alpha+1) a_0 + b_{-1}) z + ((\\alpha+2) a_1 + b_0) z^2 + \\cdots\n", "\\end{align*}\n", "we have\n", "$$\n", "(\\alpha(\\alpha+1) + z (\\alpha+1) a(z) + z^2 b(z) ) v \\equiv \n", "\\alpha \\begin{pmatrix} \n", "2 \\alpha + a_{-1} \\\\ \n", "(\\alpha+1) a_0 + b_{-1} & 2 \\alpha + a_{-1} \\\\ \n", "(\\alpha+1) a_1 + b_0 & (\\alpha+1) a_0 + b_{-1} & 2 \\alpha + a_{-1} \\\\ \n", "(\\alpha+1) a_2 + b_1 & (\\alpha+1) a_1 + b_0 & (\\alpha+1) a_0 + b_{-1} & 2 \\alpha + a_{-1} \\\\ \n", "\\vdots &\\ddots&\\ddots&\\ddots&\\ddots \\end{pmatrix}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We need to only worry about the diagonal to determine if forward substitution proceeds, in this case we have\n", "$$\n", "\\begin{pmatrix}\n", "2 \\alpha + a_{-1} \\\\\n", "\\times & 4 \\alpha + 2 + 2a_{-1} \\\\\n", "\\times & \\times &6 \\alpha + 3a_{-1} + 6 \\\\\n", "\\times & \\times & \\times & 8 \\alpha + 4a_{-1} + 12 \\\\\n", "\\times & \\times & \\times & \\times & 10 \\alpha + 5a_{-1} + 20\\\\\n", "\\vdots & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots\n", "\\end{pmatrix}\n", "$$\n", "Thus the first condition for this to give us a solution is that $2 (k+1)\\alpha + (k+1)a_{-1} + k(k+1) \\neq 0$ for all $k=1,2,\\ldots$.\n", "This can be thought of another way: note that plugging $\\alpha+k+1$ in to the indicial equation gives us:\n", "$$\n", "(\\alpha+k+1) (\\alpha+k) + a_{-1} (\\alpha+k+1) + b_{-2} = (\\alpha) (\\alpha-1) + a_{-1} \\alpha + b_{-2}+ 2 (k+1)\\alpha + (k+1)a_{-1} + k(k+1) = 2 (k+1)\\alpha + (k+1)a_{-1} + k(k+1) \n", "$$\n", "Thus forward substitution fails if and only $\\alpha+k+1$ is the second root to the indicial equation for some $k=0,1,2,\\ldots$.\n", "\n", "**Remark** If this condition is satisfied, then $v$ is analytic near the singular point, which follows by bounding the forward elimination as before, but we'll skip this for now.\n", "\n", "**Remark** If $\\alpha+k+1$ is an integer, then using the ansatz $z^{\\alpha+k+1}(1 + z v(z))$ proceeds without problem. To get the second solution, we need to introduce logarithms, but we'll skip this for now." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We now check the results of solving\n", "$$\n", "z^2 v''+ z (2(\\alpha+1) + z a(z)) v' + ( \\alpha(\\alpha+1) + z a(z) (\\alpha+1) + z^2 b(z)) v = - {\\alpha(\\alpha-1) \\over z} - a(z) \\alpha -z b(z)\n", "$$\n", "numerically.\n", "\n", "We first choose a solutino to the an indicial equation:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "α = 1.3830951894845303\n" ] }, { "data": { "text/plain": [ "2.7755575615628914e-16" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S = Taylor()\n", "z = Fun(S)\n", "D = Derivative() : S\n", "\n", "a₋₁ = -0.6\n", "b₋₂ = 0.3\n", "\n", "a = z -> a₋₁/z + 1/(2-z)\n", "ã = Fun( z -> a(z) - a₋₁/z, S)\n", "za = Fun( z -> z*a(z), S)\n", "\n", "b = z -> b₋₂/z^2 + 0.4/z + 1/(3-z)\n", "zb̃ = Fun(z -> z*b(z) - b₋₂/z, S)\n", "z²b = Fun( z -> z^2*b(z), S)\n", "\n", "α = ((1-a₋₁) + sqrt((a₋₁-1)^2-4b₋₂))/2\n", "@show α\n", "α*(α-1) + α*a₋₁ + b₋₂" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can solve for `v`, which is analytic in a neighbourhood of 0:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n", "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 600 400\">\n", "<defs>\n", " <clipPath id=\"clip00\">\n", " <rect x=\"0\" y=\"0\" width=\"600\" height=\"400\"/>\n", " </clipPath>\n", "</defs>\n", "<polygon clip-path=\"url(#clip7700)\" points=\"\n", "0,400 600,400 600,0 0,0 \n", " \" fill=\"#ffffff\" fill-opacity=\"1\"/>\n", "<defs>\n", " <clipPath id=\"clip01\">\n", " <rect x=\"120\" y=\"0\" width=\"421\" height=\"400\"/>\n", " </clipPath>\n", "</defs>\n", 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z²b)\n", "v = L \\ (-α*ã - zb̃)\n", "\n", "phaseplot(v, (-3,3), (-3,3))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Thus one solution is:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n", "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 600 400\">\n", "<defs>\n", " <clipPath id=\"clip00\">\n", " <rect x=\"0\" y=\"0\" width=\"600\" height=\"400\"/>\n", " </clipPath>\n", "</defs>\n", "<polygon clip-path=\"url(#clip7900)\" points=\"\n", "0,400 600,400 600,0 0,0 \n", " \" fill=\"#ffffff\" fill-opacity=\"1\"/>\n", "<defs>\n", " <clipPath id=\"clip01\">\n", " <rect x=\"120\" y=\"0\" width=\"421\" height=\"400\"/>\n", " </clipPath>\n", "</defs>\n", "<polygon clip-path=\"url(#clip7900)\" points=\"\n", "29.2591,375.813 580.315,375.813 580.315,11.811 29.2591,11.811 \n", " \" fill=\"#ffffff\" 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"outputs": [ { "data": { "text/plain": [ "3.3654295491028256e-12" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# dual numbers don't work to second degree so let's check numerically: \n", "uf = Fun(u, 0.1 .. 0.2)\n", "norm(uf'' + Fun(a, space(uf))*uf' + Fun(b, space(uf))*uf)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The second, linearly independent solution can be found taking the other $\\alpha$:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "α̃ = 0.21690481051546984\n" ] }, { "data": { "text/plain": [ "1.6653345369377348e-16" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "α̃ = ((1-a₋₁) - sqrt((a₋₁-1)^2-4b₋₂))/2\n", "@show α̃\n", "α̃*(α̃-1) + α̃*a₋₁ + b₋₂" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ "<?xml version=\"1.0\" 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Irregular singular points correspond to essential singularities: in this case the solution is\n", "$$\n", "u(z) = \\E^{-1/z}\n", "$$\n", "\n", "### Singular points at infinity\n", "\n", "To determine the type of singularity at $\\infty$ we do our usual change of variables $u(z) = v(1/z)$. For example, if\n", "$$\n", "{\\D u \\over \\dz} = a(z) u\n", "$$\n", "Then \n", "$$\n", "{\\D v \\over \\dz} = -z^{-2}a(z^{-1}) v\n", "$$\n", "**Definition (first order ordinary point at ∞)** $\\infty$ is an ordinary point of a first order ODE if $z^2a(z)$ is analytic at $\\infty$. \n", "\n", "**Definition (first order regular singular point at ∞)** $\\infty$ is a regular singular point of a first order ODE if $za(z) $ is analytic at $\\infty$. \n", "\n", "For second order, we see that $v$ satisfies\n", "\\begin{align*}\n", "v'(z) &= -z^{-2} u'(z^{-1}) \\\\\n", "v''(z) &=2 z^{-3} u'(z^{-1}) + z^{-4} u''(z^{-1}) =-2 z^{-1} v'(z)+ z^{-4} u''(z^{-1})\n", "\\end{align*}\n", "which gives us the ODE\n", "$$\n", "u'' + a(z) u' + b(z) u \\Rightarrow\n", "z^4v'' + (2 z^3 - a z^2) v' + b v\n", "$$\n", "\n", "\n", "**Definition (second order regular singular point at ∞)** $\\infty$ is a regular singular point of a first order ODE if $za(z)$ and $z^2 b(z)$ are analytic at $\\infty$. \n", "\n", "\n" ] } ], "metadata": { "kernelspec": { "display_name": "Julia 0.6.0", "language": "julia", "name": "julia-0.6" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "0.6.1" } }, "nbformat": 4, "nbformat_minor": 2 }